= Strong-bond limit of the alternating-spring chain
Take equal masses $m$, atomic spacing $a$, and alternating springs $K\gg\kappa>0$. The <primitive unit cell> contains a strong-bonded pair, so its length is $2a$ and the <Brillouin zone> ends at $\pm\pi/(2a)$. Expanding the exact <alternating-spring chain dispersion relation> gives
$$
\omega_-^2(q)\simeq\frac{2\kappa}{m}\sin^2(qa),\qquad
\omega_+^2(q)\simeq\frac{2K}{m}+\frac{2\kappa}{m}\cos^2(qa).
$$
The low-frequency <acoustic phonon branch> moves almost rigid dimers of mass $2m$ coupled by weak springs; its <speed of sound> tends to $a\sqrt{2\kappa/m}$. The high-frequency <optical phonon branch> describes internal dimer stretch. At the zone edge the exact frequencies are $\sqrt{2\kappa/m}$ and $\sqrt{2K/m}$.
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